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设想一个绝热房间(墙壁既不吸热也不放热),在房间某处短暂加热后,热量会从高温区域向低温区域扩散并最终趋于均匀。人们基于直觉认为,在时间足够长但尚未平衡时,房间极热或极冷点更可能出现在墙面边缘,而非房间内部——这正是美国数学家Rauch于1974年提出的“热点猜想”。

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(图片来源:华南理工大学官网)

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    设想一个绝热房间(墙壁既不吸热也不放热),在房间某处短暂加热后,热量会从高温区域向低温区域扩散并最终趋于均匀。人们基于直觉认为,在时间足够长但尚未平衡时,房间极热或极冷点更可能出现在墙面边缘,而非房间内部——这正是美国数学家Rauch于1974年提出的“热点猜想”。

    . 绝热房间“最热点”为何总在墙角. 科技导报, 2026 , 44 (2) : 11 -11 .
    . Science & Technology Review, 2026 , 44 (2) : 11 -11 .
    设想一个绝热房间(墙壁既不吸热也不放热),在房间某处短暂加热后,热量会从高温区域向低温区域扩散并最终趋于均匀。人们基于直觉认为,在时间足够长但尚未平衡时,房间极热或极冷点更可能出现在墙面边缘,而非房间内部——这正是美国数学家Rauch于1974年提出的“热点猜想”。
    华南理工大学数学学院姚若飞团队联合西安交通大学陈红斌、澳门大学桂长峰等研究人员,成功破解了“热点猜想”这一关键难题。2026年1月13日,相关研究成果发表于《Inventiones Mathematicae》。
    从数学角度来看,温度的演化遵循带有绝热(neumann)边界条件的热方程。在时间足够长的情况下,温度分布主要由“衰减最慢的首个非平凡模式”主导,即拉普拉斯算子的第二neumann特征函数(第一特征函数为常数,对应完全均温状态)。拉普拉斯算子的特征函数,就是“系统最自然、最基本的变化模式”,它规定了在给定几何形状和边界条件下,温度、振动、声压、浓度等场量,最容易以什么空间形态变化。因此,在数学家眼中,“热点猜想”可等价表述为:对于平面上的凸区域,拉普拉斯算子的第二neumann特征函数的最大值与最小值只能在区域边界处取得。
    半个多世纪以来,“热点猜想”持续吸引着国际数学界的广泛关注。菲尔兹奖得主陶哲轩、Wendelin Werner及美国艺术与科学院院士、麻省理工学院David Jerison等多位顶尖学者,围绕不同几何区域和特殊情形取得了一系列重要进展。然而,作为最基本的凸多边形之一,平面三角形虽结构简单,但因缺乏连续对称性,且无法像多边形进行拆分,因此角点带来的边界奇异性会放大到整个区域,使得扩散与振动模式对几何细节高度敏感。这种“最小却最不平衡”的结构,恰恰成为检验数学物理方法极限的典型模型。
    在这项研究中,姚若飞等系统揭示了三角形中第二neumann特征函数的几何结构,证明其内部至多存在一个鞍型临界点(热场中的“平衡节点”),而全局极值必然出现在最长边的端点。这一结果不仅解决了多个长期悬而未决的数学猜想,也表明在缺乏对称性的简单几何结构中,扩散与振动的最强响应仍受到严格的几何约束,为谱几何与工程物理问题提供了重要理论基准,如材料的结构疲劳分析、热应力集中判断等。
    (综合:《Inventiones Mathematicae》、华南理工大学官网、《中国科学报》)

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    2种不同金属材料的力学参数

    Family
    属数
    Number of
    genus
    种数
    Number of
    species
    占总种数比例
    Percentage of
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    Genus
    种数
    Number of
    species
    占总种数比例
    Percentage of total
    species (%)
    鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
    小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
    多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
    红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
    小菇属 Mycena 11 5.26
    光柄菇属 Pluteus 5 2.39
    红菇属 Russula 17 8.13
    栓菌属 Trametes 5 2.39
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